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-revision 1.1 by adcroft,
Wed Aug  8 16:15:21 2001 UTC
+revision 1.3 by adcroft,
Thu Aug  9 16:26:43 2001 UTC
 Line 3
  \section{Spatial discretization of the dynamical equations}
+ Spatial discretization is carried out using the finite volume
+ method. This amounts to a grid-point method (namely second-order
+ centered finite difference) in the fluid interior but allows
+ boundaries to intersect a regular grid allowing a more accurate
+ representation of the position of the boundary. We treat the
+ horizontal and veritical directions as seperable and thus slightly
+ differently.
+ Initialization of grid data is controlled by subroutine {\em
+ INI\_GRID} which in calls {\em INI\_VERTICAL\_GRID} to initialize the
+ vertical grid, and then either of {\em INI\_CARTESIAN\_GRID}, {\em
+ INI\_SPHERICAL\_POLAR\_GRID} or {\em INI\_CURV\-ILINEAR\_GRID} to
+ initialize the horizontal grid for cartesian, spherical-polar or
+ curvilinear coordinates respectively.
+ The reciprocals of all grid quantities are pre-calculated and this is
+ done in subroutine {\em INI\_MASKS\_ETC} which is called later by
+ subroutine {\em INITIALIZE\_FIXED}.
+ All grid descriptors are global arrays and stored in common blocks in
+ {\em GRID.h} and a generally declared as {\em \_RS}.
+ \fbox{ \begin{minipage}{4.75in}
+ {\em S/R INI\_GRID} ({\em model/src/ini\_grid.F})
+ {\em S/R INI\_MASKS\_ETC} ({\em model/src/ini\_masks\_etc.F})
+ grid data: ({\em model/inc/GRID.h})
+ \end{minipage} }
+ \subsection{The finite volume method: finite volumes versus finite difference}
+ The finite volume method is used to discretize the equations in
+ space. The expression ``finite volume'' actually has two meanings; one
+ is the method of cut or instecting boundaries (shaved or lopped cells
+ in our terminology) and the other is non-linear interpolation methods
+ that can deal with non-smooth solutions such as shocks (i.e. flux
+ limiters for advection). Both make use of the integral form of the
+ conservation laws to which the {\it weak solution} is a solution on
+ each finite volume of (sub-domain). The weak solution can be
+ constructed outof piece-wise constant elements or be
+ differentiable. The differentiable equations can not be satisfied by
+ piece-wise constant functions.
+ As an example, the 1-D constant coefficient advection-diffusion
+ equation:
+ \begin{displaymath}
+ \partial_t \theta + \partial_x ( u \theta - \kappa \partial_x \theta ) = 0
+ \end{displaymath}
+ can be discretized by integrating over finite sub-domains, i.e.
+ the lengths $\Delta x_i$:
+ \begin{displaymath}
+ \Delta x \partial_t \theta + \delta_i ( F ) = 0
+ \end{displaymath}
+ is exact if $\theta(x)$ is peice-wise constant over the interval
+ $\Delta x_i$ or more generally if $\theta_i$ is defined as the average
+ over the interval $\Delta x_i$.
+ The flux, $F_{i-1/2}$, must be approximated:
+ \begin{displaymath}
+ F = u \overline{\theta} - \frac{\kappa}{\Delta x_c} \partial_i \theta
+ \end{displaymath}
+ and this is where truncation errors can enter the solution. The
+ method for obtaining $\overline{\theta}$ is unspecified and a wide
+ range of possibilities exist including centered and upwind
+ interpolation, polynomial fits based on the the volume average
+ definitions of quantities and non-linear interpolation such as
+ flux-limiters.
+ Choosing simple centered second-order interpolation and differencing
+ recovers the same ODE's resulting from finite differencing for the
+ interior of a fluid. Differences arise at boundaries where a boundary
+ is not positioned on a regular or smoothly varying grid. This method
+ is used to represent the topography using lopped cell, see
+ \cite{Adcroft98}. Subtle difference also appear in more than one
+ dimension away from boundaries. This happens because the each
+ direction is discretized independantly in the finite difference method
+ while the integrating over finite volume implicitly treats all
+ directions simultaneously. Illustration of this is given in
+ \cite{Adcroft02}.
  \subsection{C grid staggering of variables}
  \begin{figure}
  \centerline{ \resizebox{!}{2in}{ \includegraphics{part2/cgrid3d.eps}} }
- \label{fig-cgrid3d}
  \caption{Three dimensional staggering of velocity components. This
  facilitates the natural discretization of the continuity and tracer
  equations. }
+ \label{fig:cgrid3d}
  \end{figure}
+ The basic algorithm employed for stepping forward the momentum
+ equations is based on retaining non-divergence of the flow at all
+ times. This is most naturally done if the components of flow are
+ staggered in space in the form of an Arakawa C grid \cite{Arakawa70}.
+ Fig. \ref{fig:cgrid3d} shows the components of flow ($u$,$v$,$w$)
+ staggered in space such that the zonal component falls on the
+ interface between continiuty cells in the zonal direction. Similarly
+ for the meridional and vertical directions.  The continiuty cell is
+ synonymous with tracer cells (they are one and the same).
  \subsection{Horizontal grid}
  \begin{figure}
  \centerline{ \begin{tabular}{cc}
-   \resizebox{!}{2in}{ \includegraphics{part2/hgrid-Ac.eps}}
+   \raisebox{1.5in}{a)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Ac.eps}}
- & \resizebox{!}{2in}{ \includegraphics{part2/hgrid-Az.eps}}
+ & \raisebox{1.5in}{b)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Az.eps}}
  \\
-   \resizebox{!}{2in}{ \includegraphics{part2/hgrid-Au.eps}}
+   \raisebox{1.5in}{c)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Au.eps}}
- & \resizebox{!}{2in}{ \includegraphics{part2/hgrid-Av.eps}}
+ & \raisebox{1.5in}{d)}\resizebox{!}{2in}{ \includegraphics{part2/hgrid-Av.eps}}
  \end{tabular} }
- \label{fig-hgrid}
+ \caption{
- \caption{Three dimensional staggering of velocity components. This
+ Staggering of horizontal grid descriptors (lengths and areas). The
- facilitates the natural discretization of the continuity and tracer
+ grid lines indicate the tracer cell boundaries and are the reference
- equations. }
+ grid for all panels. a) The area of a tracer cell, $A_c$, is bordered
+ by the lengths $\Delta x_g$ and $\Delta y_g$. b) The area of a
+ vorticity cell, $A_\zeta$, is bordered by the lengths $\Delta x_c$ and
+ $\Delta y_c$. c) The area of a u cell, $A_c$, is bordered by the
+ lengths $\Delta x_v$ and $\Delta y_f$. d) The area of a v cell, $A_c$,
+ is bordered by the lengths $\Delta x_f$ and $\Delta y_u$.}
+ \label{fig:hgrid}
  \end{figure}
+ The model domain is decomposed into tiles and within each tile a
+ quasi-regular grid is used. A tile is the basic unit of domain
+ decomposition for parallelization but may be used whether parallized
+ or not; see section \ref{sect:tiles} for more details. Although the
+ tiles may be patched together in an unstructured manner
+ (i.e. irregular or non-tessilating pattern), the interior of tiles is
+ a structered grid of quadrilateral cells. The horizontal coordinate
+ system is orthogonal curvilinear meaning we can not necessarily treat
+ the two horizontal directions as seperable. Instead, each cell in the
+ horizontal grid is described by the length of it's sides and it's
+ area.
+ The grid information is quite general and describes any of the
+ available coordinates systems, cartesian, spherical-polar or
+ curvilinear. All that is necessary to distinguish between the
+ coordinate systems is to initialize the grid data (discriptors)
+ appropriately.
+ In the following, we refer to the orientation of quantities on the
+ computational grid using geographic terminology such as points of the
+ compass.
+ \marginpar{Caution!}
+ This is purely for convenience but should note be confused
+ with the actual geographic orientation of model quantities.
+ Fig.~\ref{fig:hgrid}a shows the tracer cell (synonymous with the
+ continuity cell). The length of the southern edge, $\Delta x_g$,
+ western edge, $\Delta y_g$ and surface area, $A_c$, presented in the
+ vertical are stored in arrays {\bf DXg}, {\bf DYg} and {\bf rAc}.
+ \marginpar{$A_c$: {\bf rAc}}
+ \marginpar{$\Delta x_g$: {\bf DXg}}
+ \marginpar{$\Delta y_g$: {\bf DYg}}
+ The ``g'' suffix indicates that the lengths are along the defining
+ grid boundaries. The ``c'' suffix associates the quantity with the
+ cell centers. The quantities are staggered in space and the indexing
+ is such that {\bf DXg(i,j)} is positioned to the south of {\bf
+ rAc(i,j)} and {\bf DYg(i,j)} positioned to the west.
+ Fig.~\ref{fig:hgrid}b shows the vorticity cell. The length of the
+ southern edge, $\Delta x_c$, western edge, $\Delta y_c$ and surface
+ area, $A_\zeta$, presented in the vertical are stored in arrays {\bf
+ DXg}, {\bf DYg} and {\bf rAz}.
+ \marginpar{$A_\zeta$: {\bf rAz}}
+ \marginpar{$\Delta x_c$: {\bf DXc}}
+ \marginpar{$\Delta y_c$: {\bf DYc}}
+ The ``z'' suffix indicates that the lengths are measured between the
+ cell centers and the ``$\zeta$'' suffix associates points with the
+ vorticity points. The quantities are staggered in space and the
+ indexing is such that {\bf DXc(i,j)} is positioned to the north of
+ {\bf rAc(i,j)} and {\bf DYc(i,j)} positioned to the east.
+ Fig.~\ref{fig:hgrid}c shows the ``u'' or western (w) cell. The length of
+ the southern edge, $\Delta x_v$, eastern edge, $\Delta y_f$ and
+ surface area, $A_w$, presented in the vertical are stored in arrays
+ {\bf DXv}, {\bf DYf} and {\bf rAw}.
+ \marginpar{$A_w$: {\bf rAw}}
+ \marginpar{$\Delta x_v$: {\bf DXv}}
+ \marginpar{$\Delta y_f$: {\bf DYf}}
+ The ``v'' suffix indicates that the length is measured between the
+ v-points, the ``f'' suffix indicates that the length is measured
+ between the (tracer) cell faces and the ``w'' suffix associates points
+ with the u-points (w stands for west). The quantities are staggered in
+ space and the indexing is such that {\bf DXv(i,j)} is positioned to
+ the south of {\bf rAw(i,j)} and {\bf DYf(i,j)} positioned to the east.
+ Fig.~\ref{fig:hgrid}d shows the ``v'' or southern (s) cell. The length of
+ the northern edge, $\Delta x_f$, western edge, $\Delta y_u$ and
+ surface area, $A_s$, presented in the vertical are stored in arrays
+ {\bf DXf}, {\bf DYu} and {\bf rAs}.
+ \marginpar{$A_s$: {\bf rAs}}
+ \marginpar{$\Delta x_f$: {\bf DXf}}
+ \marginpar{$\Delta y_u$: {\bf DYu}}
+ The ``u'' suffix indicates that the length is measured between the
+ u-points, the ``f'' suffix indicates that the length is measured
+ between the (tracer) cell faces and the ``s'' suffix associates points
+ with the v-points (s stands for south). The quantities are staggered
+ in space and the indexing is such that {\bf DXf(i,j)} is positioned to
+ the north of {\bf rAs(i,j)} and {\bf DYu(i,j)} positioned to the west.
+ \fbox{ \begin{minipage}{4.75in}
+ {\em S/R INI\_CARTESIAN\_GRID} ({\em
+ model/src/ini\_cartesian\_grid.F})
+ {\em S/R INI\_SPHERICAL\_POLAR\_GRID} ({\em
+ model/src/ini\_spherical\_polar\_grid.F})
+ {\em S/R INI\_CURVILINEAR\_GRID} ({\em
+ model/src/ini\_curvilinear\_grid.F})
+ $A_c$, $A_\zeta$, $A_w$, $A_s$: {\bf rAc}, {\bf rAz}, {\bf rAw}, {\bf rAs}
+ ({\em GRID.h})
+ $\Delta x_g$, $\Delta y_g$: {\bf DXg}, {\bf DYg} ({\em GRID.h})
+ $\Delta x_c$, $\Delta y_c$: {\bf DXc}, {\bf DYc} ({\em GRID.h})
+ $\Delta x_f$, $\Delta y_f$: {\bf DXf}, {\bf DYf} ({\em GRID.h})
+ $\Delta x_v$, $\Delta y_u$: {\bf DXv}, {\bf DYu} ({\em GRID.h})
+ \end{minipage} }
+ \subsubsection{Reciprocals of horizontal grid descriptors}
+ %\marginpar{$A_c^{-1}$: {\bf RECIP\_rAc}}
+ %\marginpar{$A_\zeta^{-1}$: {\bf RECIP\_rAz}}
+ %\marginpar{$A_w^{-1}$: {\bf RECIP\_rAw}}
+ %\marginpar{$A_s^{-1}$: {\bf RECIP\_rAs}}
+ Lengths and areas appear in the denominator of expressions as much as
+ in the numerator. For efficiency and portability, we pre-calculate the
+ reciprocal of the horizontal grid quantities so that in-line divisions
+ can be avoided.
+ %\marginpar{$\Delta x_g^{-1}$: {\bf RECIP\_DXg}}
+ %\marginpar{$\Delta y_g^{-1}$: {\bf RECIP\_DYg}}
+ %\marginpar{$\Delta x_c^{-1}$: {\bf RECIP\_DXc}}
+ %\marginpar{$\Delta y_c^{-1}$: {\bf RECIP\_DYc}}
+ %\marginpar{$\Delta x_f^{-1}$: {\bf RECIP\_DXf}}
+ %\marginpar{$\Delta y_f^{-1}$: {\bf RECIP\_DYf}}
+ %\marginpar{$\Delta x_v^{-1}$: {\bf RECIP\_DXv}}
+ %\marginpar{$\Delta y_u^{-1}$: {\bf RECIP\_DYu}}
+ For each grid descriptor (array) there is a reciprocal named using the
+ prefix {\bf RECIP\_}. This doubles the amount of storage in {\em
+ GRID.h} but they are all only 2-D descriptors.
+ \fbox{ \begin{minipage}{4.75in}
+ {\em S/R INI\_MASKS\_ETC} ({\em
+ model/src/ini\_masks\_etc.F})
+ $A_c^{-1}$: {\bf RECIP\_Ac} ({\em GRID.h})
+ $A_\zeta^{-1}$: {\bf RECIP\_Az} ({\em GRID.h})
+ $A_w^{-1}$: {\bf RECIP\_Aw} ({\em GRID.h})
+ $A_s^{-1}$: {\bf RECIP\_As} ({\em GRID.h})
+ $\Delta x_g^{-1}$, $\Delta y_g^{-1}$: {\bf RECIP\_DXg}, {\bf RECIP\_DYg} ({\em GRID.h})
+ $\Delta x_c^{-1}$, $\Delta y_c^{-1}$: {\bf RECIP\_DXc}, {\bf RECIP\_DYc} ({\em GRID.h})
+ $\Delta x_f^{-1}$, $\Delta y_f^{-1}$: {\bf RECIP\_DXf}, {\bf RECIP\_DYf} ({\em GRID.h})
+ $\Delta x_v^{-1}$, $\Delta y_u^{-1}$: {\bf RECIP\_DXv}, {\bf RECIP\_DYu} ({\em GRID.h})
+ \end{minipage} }
+ \subsubsection{Cartesian coordinates}
+ Cartesian coordinates are selected when the logical flag {\bf
+ using\-Cartes\-ianGrid} in namelist {\em PARM04} is set to true. The grid
+ spacing can be set to uniform via scalars {\bf dXspacing} and {\bf
+ dYspacing} in namelist {\em PARM04} or to variable resolution by the
+ vectors {\bf DELX} and {\bf DELY}. Units are normally
+ meters. Non-dimensional coordinates can be used by interpretting the
+ gravitational constant as the Rayleigh number.
+ \subsubsection{Spherical-polar coordinates}
+ Spherical coordinates are selected when the logical flag {\bf
+ using\-Spherical\-PolarGrid} in namelist {\em PARM04} is set to true. The
+ grid spacing can be set to uniform via scalars {\bf dXspacing} and
+ {\bf dYspacing} in namelist {\em PARM04} or to variable resolution by
+ the vectors {\bf DELX} and {\bf DELY}. Units of these namelist
+ variables are alway degrees. The horizontal grid descriptors are
+ calculated from these namelist variables have units of meters.
+ \subsubsection{Curvilinear coordinates}
+ Curvilinear coordinates are selected when the logical flag {\bf
+ using\-Curvil\-inear\-Grid} in namelist {\em PARM04} is set to true. The
+ grid spacing can not be set via the namelist. Instead, the grid
+ descriptors are read from data files, one for each descriptor. As for
+ other grids, the horizontal grid descriptors have units of meters.
  \subsection{Vertical grid}
  \begin{figure}
  \centerline{ \begin{tabular}{cc}
-   \raisebox{4in}{a)}
+   \raisebox{4in}{a)} \resizebox{!}{4in}{
-   \resizebox{!}{4in}{ \includegraphics{part2/vgrid-cellcentered.eps}}
+   \includegraphics{part2/vgrid-cellcentered.eps}} & \raisebox{4in}{b)}
- &
+   \resizebox{!}{4in}{ \includegraphics{part2/vgrid-accurate.eps}}
-  \raisebox{4in}{b)}
-  \resizebox{!}{4in}{ \includegraphics{part2/vgrid-accurate.eps}}
  \end{tabular} }
- \label{fig-vgrid}
  \caption{Two versions of the vertical grid. a) The cell centered
  approach where the interface depths are specified and the tracer
  points centered in between the interfaces. b) The interface centered
  approach where tracer levels are specified and the w-interfaces are
  centered in between.}
+ \label{fig:vgrid}
+ \end{figure}
+ As for the horizontal grid, we use the suffixes ``c'' and ``f'' to
+ indicates faces and centers. Fig.~\ref{fig:vgrid}a shows the default
+ vertical grid used by the model.
+ \marginpar{$\Delta r_f$: {\bf DRf}}
+ \marginpar{$\Delta r_c$: {\bf DRc}}
+ $\Delta r_f$ is the difference in $r$
+ (vertical coordinate) between the faces (i.e. $\Delta r_f \equiv -
+ \delta_k r$ where the minus sign appears due to the convention that the
+ surface layer has index $k=1$.).
+ The vertical grid is calculated in subroutine {\em
+ INI\_VERTICAL\_GRID} and specified via the vector {\bf DELR} in
+ namelist {\em PARM04}. The units of ``r'' are either meters or Pascals
+ depending on the isomorphism being used which in turn is dependent
+ only on the choise of equation of state.
+ There are alternative namelist vectors {\bf DELZ} and {\bf DELP} which
+ dictate whether z- or
+ \marginpar{Caution!}
+ p- coordinates are to be used but we intend to
+ phase this out since they are redundant.
+ The reciprocals $\Delta r_f^{-1}$ and $\Delta r_c^{-1}$ are
+ pre-calculated (also in subroutine {\em INI\_VERTICAL\_GRID}). All
+ vertical grid descriptors are stored in common blocks in {\em GRID.h}.
+ The above grid (Fig.~\ref{fig:vgrid}a) is known as the cell centered
+ approach because the tracer points are at cell centers; the cell
+ centers are mid-way between the cell interfaces. An alternative, the
+ vertex or interface centered approach, is shown in
+ Fig.~\ref{fig:vgrid}b. Here, the interior interfaces are positioned
+ mid-way between the tracer nodes (no longer cell centers). This
+ approach is formally more accurate for evaluation of hydrostatic
+ pressure and vertical advection but historically the cell centered
+ approach has been used. An alternative form of subroutine {\em
+ INI\_VERTICAL\_GRID} is used to select the interface centered approach
+ but no run time option is currently available.
+ \fbox{ \begin{minipage}{4.75in}
+ {\em S/R INI\_VERTICAL\_GRID} ({\em
+ model/src/ini\_vertical\_grid.F})
+ $\Delta r_f$: {\bf DRf} ({\em GRID.h})
+ $\Delta r_c$: {\bf DRc} ({\em GRID.h})
+ $\Delta r_f^{-1}$: {\bf RECIP\_DRf} ({\em GRID.h})
+ $\Delta r_c^{-1}$: {\bf RECIP\_DRc} ({\em GRID.h})
+ \end{minipage} }
+ \subsection{Topography: partially filled cells}
+ \begin{figure}
+ \centerline{
+ \resizebox{4.5in}{!}{\includegraphics{part2/vgrid-xz.eps}}
+ }
+ \caption{
+ A schematic of the x-r plane showing the location of the
+ non-dimensional fractions $h_c$ and $h_w$. The physical thickness of a
+ tracer cell is given by $h_c(i,j,k) \Delta r_f(k)$ and the physical
+ thickness of the open side is given by $h_w(i,j,k) \Delta r_f(k)$.}
+ \label{fig:hfacs}
  \end{figure}
+ \cite{Adcroft97} presented two alternatives to the step-wise finite
+ difference representation of topography. The method is known to the
+ engineering community as {\em intersecting boundary method}. It
+ involves allowing the boundary to intersect a grid of cells thereby
+ modifying the shape of those cells intersected. We suggested allowing
+ the topgoraphy to take on a peice-wise linear representation (shaved
+ cells) or a simpler piecewise constant representaion (partial step).
+ Both show dramatic improvements in solution compared to the
+ traditional full step representation, the piece-wise linear being the
+ best. However, the storage requirements are excessive so the simpler
+ piece-wise constant or partial-step method is all that is currently
+ supported.
+ Fig.~\ref{fig:hfacs} shows a schematic of the x-r plane indicating how
+ the thickness of a level is determined at tracer and u points.
+ \marginpar{$h_c$: {\bf hFacC}}
+ \marginpar{$h_w$: {\bf hFacW}}
+ \marginpar{$h_s$: {\bf hFacS}}
+ The physical thickness of a tracer cell is given by $h_c(i,j,k) \Delta
+ r_f(k)$ and the physical thickness of the open side is given by
+ $h_w(i,j,k) \Delta r_f(k)$. Three 3-D discriptors $h_c$, $h_w$ and
+ $h_s$ are used to describe the geometry: {\bf hFacC}, {\bf hFacW} and
+ {\bf hFacS} respectively. These are calculated in subroutine {\em
+ INI\_MASKS\_ETC} along with there reciprocals {\bf RECIP\_hFacC}, {\bf
+ RECIP\_hFacW} and {\bf RECIP\_hFacS}.
+ The non-dimensional fractions (or h-facs as we call them) are
+ calculated from the model depth array and then processed to avoid tiny
+ volumes. The rule is that if a fraction is less than {\bf hFacMin}
+ then it is rounded to the nearer of $0$ or {\bf hFacMin} or if the
+ physical thickness is less than {\bf hFacMinDr} then it is similarly
+ rounded. The larger of the two methods is used when there is a
+ conflict. By setting {\bf hFacMinDr} equal to or larger than the
+ thinnest nominal layers, $\min{(\Delta z_f)}$, but setting {\bf
+ hFacMin} to some small fraction then the model will only lop thick
+ layers but retain stability based on the thinnest unlopped thickness;
+ $\min{(\Delta z_f,\mbox{\bf hFacMinDr})}$.
+ \fbox{ \begin{minipage}{4.75in}
+ {\em S/R INI\_MASKS\_ETC} ({\em model/src/ini\_masks\_etc.F})
+ $h_c$: {\bf hFacC} ({\em GRID.h})
+ $h_w$: {\bf hFacW} ({\em GRID.h})
+ $h_s$: {\bf hFacS} ({\em GRID.h})
+ $h_c^{-1}$: {\bf RECIP\_hFacC} ({\em GRID.h})
+ $h_w^{-1}$: {\bf RECIP\_hFacW} ({\em GRID.h})
+ $h_s^{-1}$: {\bf RECIP\_hFacS} ({\em GRID.h})
+ \end{minipage} }
  \subsection{Continuity and horizontal pressure gradient terms}
-Line 59 
 continuity equation which can be summari
+Line 457 
 continuity equation which can be summari
  \delta_i \Delta y_g \Delta r_f h_w u +
  \delta_j \Delta x_g \Delta r_f h_s v +
  \delta_k {\cal A}_c w & = & {\cal A}_c \delta_k (P-E)_{r=0}
+ \label{eq:discrete-continuity}
  \end{eqnarray}
  where the continuity equation has been most naturally discretized by
  staggering the three components of velocity as shown in
-Line 154 
 These terms are calculated in routines c
+Line 553 
 These terms are calculated in routines c
  CALC\_MOM\_RHS} a collected into the global arrays {\bf Gu}, {\bf Gv},
  and {\bf Gw}.
- \fbox{ \begin{minipage}{4.25in}
+ \fbox{ \begin{minipage}{4.75in}
  {\em S/R CALC\_MOM\_RHS} ({\em pkg/mom\_fluxform/calc\_mom\_rhs.F})
  $G_u$: {\bf Gu} ({\em DYNVARS.h})
-Line 199 
 conservation of second moments, derived
+Line 598 
 conservation of second moments, derived
  $u^2$ and $v^2$ and $w^2$ so that advection of momentum correctly
  conserves kinetic energy.
- \fbox{ \begin{minipage}{4.25in}
+ \fbox{ \begin{minipage}{4.75in}
  {\em S/R MOM\_U\_ADV\_UU} ({\em mom\_u\_adv\_uu.F})
  {\em S/R MOM\_U\_ADV\_VU} ({\em mom\_u\_adv\_vu.F})
-Line 263 
 compatibility testing (set run-time logi
+Line 662 
 compatibility testing (set run-time logi
  useNonconservingCoriolis} to {\em true} which otherwise defaults to
  {\em false}).
- \fbox{ \begin{minipage}{4.25in}
+ \fbox{ \begin{minipage}{4.75in}
  {\em S/R MOM\_CDSCHEME} ({\em mom\_cdscheme.F})
  {\em S/R MOM\_U\_CORIOLIS} ({\em mom\_u\_coriolis.F})
-Line 304 
 G_v^{metric} & = & \frac{ \overline{u}^{
+Line 703 
 G_v^{metric} & = & \frac{ \overline{u}^{
  where $\tan{\phi}$ is evaluated at the $u$ and $v$ points
  respectively.
- \fbox{ \begin{minipage}{4.25in}
+ \fbox{ \begin{minipage}{4.75in}
  {\em S/R MOM\_U\_METRIC\_SPHERE} ({\em mom\_u\_metric\_sphere.F})
  {\em S/R MOM\_V\_METRIC\_SPHERE} ({\em mom\_v\_metric\_sphere.F})
-Line 339 
 G_w^{metric} & = &
+Line 738 
 G_w^{metric} & = &
    \frac{1}{a} ( {\overline{u}^{ik}}^2 + {\overline{v}^{jk}}^2 )
  \end{eqnarray}
- \fbox{ \begin{minipage}{4.25in}
+ \fbox{ \begin{minipage}{4.75in}
  {\em S/R MOM\_U\_METRIC\_NH} ({\em mom\_u\_metric\_nh.F})
  {\em S/R MOM\_V\_METRIC\_NH} ({\em mom\_v\_metric\_nh.F})
-Line 391 
 The Laplacian viscosity coefficient, $A_
+Line 790 
 The Laplacian viscosity coefficient, $A_
  of $m^2 s^{-1}$. The bi-harmonic viscosity coefficient, $A_4$ ({\bf
  viscA4}), has units of $m^4 s^{-1}$.
- \fbox{ \begin{minipage}{4.25in}
+ \fbox{ \begin{minipage}{4.75in}
  {\em S/R MOM\_U\_XVISCFLUX} ({\em mom\_u\_xviscflux.F})
  {\em S/R MOM\_U\_YVISCFLUX} ({\em mom\_u\_yviscflux.F})
-Line 434 
 In fact, the above discretization is not
+Line 833 
 In fact, the above discretization is not
  assumes that the bathymetry at velocity points is deeper than at
  neighbouring vorticity points, e.g. $1-h_w < 1-h_\zeta$
- \fbox{ \begin{minipage}{4.25in}
+ \fbox{ \begin{minipage}{4.75in}
  {\em S/R MOM\_U\_SIDEDRAG} ({\em mom\_u\_sidedrag.F})
  {\em S/R MOM\_V\_SIDEDRAG} ({\em mom\_v\_sidedrag.F})
-Line 471 
 is even less consistent than for the hyd
+Line 870 
 is even less consistent than for the hyd
  \cite{Waojz}). It is well known how to do this properly (see Griffies
  \cite{Griffies}) and is on the list of to-do's.
- \fbox{ \begin{minipage}{4.25in}
+ \fbox{ \begin{minipage}{4.75in}
  {\em S/R MOM\_U\_RVISCLFUX} ({\em mom\_u\_rviscflux.F})
  {\em S/R MOM\_V\_RVISCLFUX} ({\em mom\_v\_rviscflux.F})
-Line 509 
 $r_b$ ({\bf bottomDragLinear}) has units
+Line 908 
 $r_b$ ({\bf bottomDragLinear}) has units
  of the order 0.0002 $m s^{-1}$. $C_d$ ({\bf bottomDragQuadratic}) is
  dimensionless with typical values in the range 0.001--0.003.
- \fbox{ \begin{minipage}{4.25in}
+ \fbox{ \begin{minipage}{4.75in}
  {\em S/R MOM\_U\_BOTTOMDRAG} ({\em mom\_u\_bottomdrag.F})
  {\em S/R MOM\_V\_BOTTOMDRAG} ({\em mom\_v\_bottomdrag.F})
-Line 609 
 G_w & = & G_w^{fu} + G_w^{\zeta_1 v} + G
+Line 1008 
 G_w & = & G_w^{fu} + G_w^{\zeta_1 v} + G
  + G_w^{h-dissip} + G_w^{v-dissip}
  \end{eqnarray}
- \fbox{ \begin{minipage}{4.25in}
+ \fbox{ \begin{minipage}{4.75in}
  {\em S/R CALC\_MOM\_RHS} ({\em pkg/mom\_vecinv/calc\_mom\_rhs.F})
  $G_u$: {\bf Gu} ({\em DYNVARS.h})
-Line 634 
 Relative vorticity is defined:
+Line 1033 
 Relative vorticity is defined:
  where ${\cal A}_\zeta$ is the area of the vorticity cell presented in
  the vertical and $\Gamma$ is the circulation about that cell.
- \fbox{ \begin{minipage}{4.25in}
+ \fbox{ \begin{minipage}{4.75in}
  {\em S/R MOM\_VI\_CALC\_RELVORT3} ({\em mom\_vi\_calc\_relvort3.F})
  $\zeta_3$: {\bf vort3} (local to {\em calc\_mom\_rhs.F})
-Line 649 
 KE = \frac{1}{2} ( \overline{ u^2 }^i +
+Line 1048 
 KE = \frac{1}{2} ( \overline{ u^2 }^i +
  + \epsilon_{nh} \overline{ w^2 }^k )
  \end{equation}
- \fbox{ \begin{minipage}{4.25in}
+ \fbox{ \begin{minipage}{4.75in}
  {\em S/R MOM\_VI\_CALC\_KE} ({\em mom\_vi\_calc\_ke.F})
  $KE$: {\bf KE} (local to {\em calc\_mom\_rhs.F})
-Line 705 
 advection of absolute vorticity. Current
+Line 1104 
 advection of absolute vorticity. Current
  relative/absolute vorticity, centered/upwind/high order advection is
  available only through commented subroutine calls.
- \fbox{ \begin{minipage}{4.25in}
+ \fbox{ \begin{minipage}{4.75in}
  {\em S/R MOM\_VI\_CORIOLIS} ({\em mom\_vi\_coriolis.F})
  {\em S/R MOM\_VI\_U\_CORIOLIS} ({\em mom\_vi\_u\_coriolis.F})
-Line 736 
 G_v^{\zeta_1 w} & = &
+Line 1135 
 G_v^{\zeta_1 w} & = &
  }^k
  \end{eqnarray}
- \fbox{ \begin{minipage}{4.25in}
+ \fbox{ \begin{minipage}{4.75in}
  {\em S/R MOM\_VI\_U\_VERTSHEAR} ({\em mom\_vi\_u\_vertshear.F})
  {\em S/R MOM\_VI\_V\_VERTSHEAR} ({\em mom\_vi\_v\_vertshear.F})
-Line 759 
 G_v^{\partial_y B} & = &
+Line 1158 
 G_v^{\partial_y B} & = &
  %\frac{1}{\Delta r_c} h_c \delta_k ( \phi' + KE )
  \end{eqnarray}
- \fbox{ \begin{minipage}{4.25in}
+ \fbox{ \begin{minipage}{4.75in}
  {\em S/R MOM\_VI\_U\_GRAD\_KE} ({\em mom\_vi\_u\_grad\_ke.F})
  {\em S/R MOM\_VI\_V\_GRAD\_KE} ({\em mom\_vi\_v\_grad\_ke.F})
-Line 782 
 D = \frac{1}{{\cal A}_c h_c} (
+Line 1181 
 D = \frac{1}{{\cal A}_c h_c} (
  + \delta_j \Delta x_g h_s v )
  \end{equation}
- \fbox{ \begin{minipage}{4.25in}
+ \fbox{ \begin{minipage}{4.75in}
  {\em S/R MOM\_VI\_CALC\_HDIV} ({\em mom\_vi\_calc\_hdiv.F})
  $D$: {\bf hDiv} (local to {\em calc\_mom\_rhs.F})
-Line 813 
 D^* & = & \frac{1}{{\cal A}_c h_c} (
+Line 1212 
 D^* & = & \frac{1}{{\cal A}_c h_c} (
  - \delta_j \Delta x_c \nabla^2 u )
  \end{eqnarray}
- \fbox{ \begin{minipage}{4.25in}
+ \fbox{ \begin{minipage}{4.75in}
  {\em S/R MOM\_VI\_HDISSIP} ({\em mom\_vi\_hdissip.F})
  $G_u^{h-dissip}$: {\bf uDiss} (local to {\em calc\_mom\_rhs.F})
-Line 839 
 In the interior the vertical stresses ar
+Line 1238 
 In the interior the vertical stresses ar
  \tau_{23} & = & A_v \frac{1}{\Delta r_c} \delta_k v
  \end{eqnarray}
- \fbox{ \begin{minipage}{4.25in}
+ \fbox{ \begin{minipage}{4.75in}
  {\em S/R MOM\_U\_RVISCLFUX} ({\em mom\_u\_rviscflux.F})
  {\em S/R MOM\_V\_RVISCLFUX} ({\em mom\_v\_rviscflux.F})

 Legend:



Removed from v.1.1
 


changed lines


 
Added in v.1.3
 Legend:



Removed from v.1.1
 


changed lines


 
Added in v.1.3
-Removed from v.1.1
+Added in v.1.3

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